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One intriguing property of deep neural networks (DNNs) is their inherent vulnerability to backdoor attacks -- a trojan model responds to trigger-embedded inputs in a highly predictable manner while functioning normally otherwise. Despite the plethora of prior work on DNNs for continuous data (e.g., images), the vulnerability of graph neural networks (GNNs) for discrete-structured data (e.g., graphs) is largely unexplored, which is highly concerning given their increasing use in security-sensitive domains. To bridge this gap, we present GTA, the first backdoor attack on GNNs. Compared with prior work, GTA departs in significant ways: graph-oriented -- it defines triggers as specific subgraphs, including both topological structures and descriptive features, entailing a large design spectrum for the adversary; input-tailored -- it dynamically adapts triggers to individual graphs, thereby optimizing both attack effectiveness and evasiveness; downstream model-agnostic -- it can be readily launched without knowledge regarding downstream models or fine-tuning strategies; and attack-extensible -- it can be instantiated for both transductive (e.g., node classification) and inductive (e.g., graph classification) tasks, constituting severe threats for a range of security-critical applications. Through extensive evaluation using benchmark datasets and state-of-the-art models, we demonstrate the effectiveness of GTA. We further provide analytical justification for its effectiveness and discuss potential countermeasures, pointing to several promising research directions.